A generalization of cancellative dimer algebras to hyperbolic surfaces

We study a new class of path algebras with relations on surfaces, called ‘geodesic ghor algebras’. These algebras generalize cancellative dimer algebras on a torus to higher genus surfaces, where the relations come from perfect matchings rather than a potential. Although cancellative dimer algebras on a torus are noncommutative crepant resolutions, the center of any dimer algebra on a higher genus surface is just the polynomial ring in one variable, and so the center and surface are unrelated. In contrast, we establish a rich interplay between the central geometry of geodesic ghor algebras and the topology of the surface in which they are embedded. Furthermore, we show that the localizations of these algebras over the noetherian locus are endomorphism rings of modules over their centers.